In 6-dimensional geometry, the **1**_{22} polytope is a uniform polytope, constructed from the E_{6} group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V_{72} (for its 72 vertices).

Its Coxeter symbol is **1**_{22}, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are two rectifications of the 1_{22}, construcated by positions points on the elements of 1_{22}. The **rectified 1**_{22} is constructed by points at the mid-edges of the **1**_{22}. The **birectified 1**_{22} is constructed by points at the triangle face centers of the **1**_{22}.

These polytopes are from a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

The 1_22 polytope contains 72 vertices, and 54 5-demicubic facets. It has a birectified 5-simplex vertex figure. Its 72 vertices represent the root vectors of the simple Lie group E_{6}.

**Pentacontatetra-peton** (Acronym Mo) - 54-facetted polypeton (Jonathan Bowers)
It is created by a Wythoff construction upon a set of 6 hyperplane mirrors in 6-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram, .

Removing the node on either of 2-length branches leaves the 5-demicube, 1_{31}, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 5-simplex, 0_{22}, .

The regular complex polyhedron _{3}{3}_{3}{4}_{2}, , in
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2
has a real representation as the *1*_{22} polytope in 4-dimensional space. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces. Its complex reflection group is _{3}[3]_{3}[4]_{2}, order 1296. It has a half-symmetry quasiregular construction as , as a rectification of the Hessian polyhedron, .

Along with the semiregular polytope, **2**_{21}, it is also one of a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

The **1**_{22} is related to the 24-cell by a geometric folding E6 → F4 of Coxeter-Dynkin diagrams, E6 corresponding to **1**_{22} in 6 dimensions, F4 to the 24-cell in 4 dimensions. This can be seen in the Coxeter plane projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the 1_{22}.

This polytope is the vertex figure for a uniform tessellation of 6-dimensional space, **2**_{22}, .

The **rectified 1**_{22} polytope (also called **0**_{221}) can tessellate 6-dimensional space as the Voronoi cell of the E6* honeycomb lattice (dual of E6 lattice).

Birectified 2_{21} polytope
Rectified pentacontatetrapeton (acronym *Ram*) - rectified 54-facetted polypeton (Jonathan Bowers)

Its construction is based on the E_{6} group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .

Removing the ring on the short branch leaves the birectified 5-simplex, .

Removing the ring on the either 2-length branch leaves the birectified 5-orthoplex in its alternated form: **t**_{2}(2_{11}), .

The vertex figure is determined by removing the ringed node and ringing the neighboring ring. This makes 3-3 duoprism prism, {3}×{3}×{}, .

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

Truncated 1_{22} polytope
Its construction is based on the E_{6} group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

Bicantellated 2_{21}
Birectified pentacontitetrapeton (barm) (Jonathan Bowers)

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

Tricantellated 2_{21}
Trirectified pentacontitetrapeton (trim) (Jonathan Bowers)